One can verify that \(\forall g \in \mathbb {Z}^d, \hat{f}(g \cdot v,e^{i k}) = e^{i g} \hat{f}(v,e^{i k})\) .
This means that the entire function \(\hat{f}\)  may be recovered from its restriction to \(W \times \mathbb {T}^d\) , hence from now on we will view \(\hat{f}\)  as a function on this restricted domain.
The following key theorem is a consequence of standard techniques from Fourier analysis. For a more detailed exposition see [1]}.
